On the embedding dimension of 2-torsion lens spaces

TL;DR

This paper establishes new lower bounds on the Euclidean embedding dimensions of 2-torsion lens spaces using advanced cobordism theories, and provides inductive constructions for embeddings close to optimal dimensions.

Contribution

It introduces novel non-embedding results for 2-torsion lens spaces based on $ku$- and $BP$-theoretic cobordism obstructions, and offers inductive methods for constructing embeddings.

Findings

Proves non-embeddability of certain lens spaces in specific Euclidean dimensions.

Provides inductive constructions for embeddings of 2^e-torsion lens spaces.

Embeddings achieved are within one dimension of the theoretical optimum.

Abstract

Using the $ku$- and $BP$-theoretic versions of Astey's cobordism obstruction for the existence of smooth Euclidean embeddings of stably almost complex manifolds, we prove that, for $e$ greater than or equal to $\alpha(n)$--the number of ones in the dyadic expansion of $n$--, the ($2n+1$)-dimensional $2^e$-torsion lens space cannot be embedded in Euclidean space of dimension $4n-2\alpha(n)+1$. A slightly restricted version of this fact holds for $e<\alpha(n)$. We also give an inductive construction of Euclidean embeddings for $2^e$-torsion lens spaces. Some of our best embeddings are within one dimension of being optimal.